The $r$-value: evaluating stability with respect to distributional shifts

Common statistical measures of uncertainty like $p$-values and confidence intervals quantify the uncertainty due to sampling, that is, the uncertainty due to not observing the full population. In practice, populations change between locations and across time. This makes it difficult to gather knowledge that transfers across data sets. We propose a measure of uncertainty that quantifies the distributional uncertainty of a statistical estimand with respect to Kullback-Liebler divergence, that is, the sensitivity of the parameter under general distributional perturbations within a Kullback-Liebler divergence ball. If the signal-to-noise ratio is small, distributional uncertainty is a monotonous transformation of the signal-to-noise ratio. In general, however, it is a different concept and corresponds to a different research question. Further, we propose measures to estimate the stability of parameters with respect to directional or variable-specific shifts. We also demonstrate how the measure of distributional uncertainty can be used to prioritize data collection for better estimation of statistical parameters under shifted distribution. We evaluate the performance of the proposed measure in simulations and real data and show that it can elucidate the distributional (in-)stability of an estimator with respect to certain shifts and give more accurate estimates of parameters under shifted distribution only requiring to collect limited information from the shifted distribution.